TOWARD AN UNDERSTANDING OF GRB PROMPT EMISSION MECHANISM. I. THE ORIGIN OF SPECTRAL LAGS

The physical picture adopted in our numerical model is simple and straightforward to understand: a single relativistic spherical shell expands radially with the bulk Lorentz factor Γ, continuously emitting photons from all locations in the shell, with an isotropic angular distribution of the emitted power in its co-moving fluid frame. The shape of the photon spectrum in the co-moving frame is delineated by a functional form (Uhm & Zhang 2015a),

$$\begin{eqnarray}&&H(x)\quad {\rm{with}}\quad x={\nu }^{\prime }/{\nu }_{{\rm{ch}}}^{\prime }.\end{eqnarray} \tag{ 1 }$$

The H(x) function is allowed to have an arbitrary shape as a function of the photon frequency ${\nu }^{\prime }$ (measured in the co-moving frame), and is located at a characteristic frequency ${\nu }_{{\rm{ch}}}^{\prime }$. The radiating electrons are distributed uniformly in the shell, and the number of the electrons N is assumed to increase at an injection rate of ${R}_{{\rm{inj}}}\equiv {dN}/{{dt}}^{\prime }$ from an initial value of N = 0. Here, ${t}^{\prime }$ is the time, measured in the co-moving frame.

The luminosity distance to the GRB site is calculated by adopting the standard flat ΛCDM universe with the parameters ${H}_{0}=71$ km ${{\rm{s}}}^{-1}$ ${{\rm{Mpc}}}^{-1},{{\rm{\Omega }}}_{{\rm{m}}}=0.27$, and ${{\rm{\Omega }}}_{{\rm{\Lambda }}}=0.73$. The redshift plays only a global role in shaping the observed spectral flux (Uhm & Zhang 2015a), and thus we assume a typical value of z = 1 in all of the numerical models presented in this paper.

The prompt gamma-rays of GRBs are not emitted from the explosion center, but rather at a certain radius far from the central engine. Hence, we consider the emission of the spherical shell to be turned on at radius ${r}_{{\rm{on}}}$ (and at the lab-frame time ${t}_{{\rm{on}}}$), such that the observer time ${t}_{{\rm{obs}}}$ is set equal to zero when the first photons from this radius ${r}_{{\rm{on}}}$ trigger the detector on Earth. Then, a photon emitted from the shell at a later lab time t ($\gt {t}_{{\rm{on}}}$) when the shell has radius r ($\gt {r}_{{\rm{on}}}$) will be received by the observer at

$$\begin{eqnarray}{t}_{{\rm{obs}}} & = & \displaystyle \frac{1}{c}[{r}_{{\rm{on}}}+c(t-{t}_{{\rm{on}}})-r\mathrm{cos}\theta ](1+z)\\ & = & \left[\left(t-\displaystyle \frac{r}{c}\mathrm{cos}\theta \right)-\left({t}_{{\rm{on}}}-\displaystyle \frac{{r}_{{\rm{on}}}}{c}\right)\right](1+z),\end{eqnarray} \tag{ 2 }$$

where c is the speed of light and the angle θ measures the latitude of the photon's emission location with respect to the observer's line of sight. Note that the observer time ${t}_{{\rm{obs}}}$ is practically independent of ${t}_{{\rm{on}}}$ since we calculate the time t as $t={t}_{{\rm{on}}}+{\int }_{{r}_{{\rm{on}}}}{dr}/(c\beta )$.

We assign a spectral power ${P}_{0}^{\prime}$ to each electron in the shell and calculate the observed spectral flux, ${F}_{{\nu }_{{\rm{obs}}}}^{\;\mathrm{obs}}$, in terms of the observer time ${t}_{{\rm{obs}}}$ and the observed frequency ${\nu }_{{\rm{obs}}}$, while fully taking into account the high-latitude curvature effect of the spherical shell, which includes the Doppler boosting from the co-moving frame to the observer frame for each latitude and the integration over the equal-arrival time surface of each observer time (Uhm & Zhang 2015a).

We begin our calculations at radius ${r}_{{\rm{on}}}$ and turn off the emission of the shell at radius ${r}_{{\rm{off}}}$. As for the shape of the photon spectrum in the co-moving frame, we use a curved spectrum such as the Band function (Band et al. 1993) with the parameters ${\alpha }_{{\rm{B}}}$ (the low-energy photon index) and ${\beta }_{{\rm{B}}}$ (the high-energy photon index). If the photon spectrum is not curved (say, a power-law shape), then the light curves at different energies will always behave in the same way, rising and/or falling simultaneously, which then leaves no possibility of producing a spectral lag.

Our calculations are not attached to a particular theoretical model of prompt emission. In the following, we adopt a general approach with the goal of identifying the physical conditions that are required by the observations.

First, we note that GRB spectral lags sample the broad pulses in the light curves, which are the "slow components," in contrast to the fast-variability components that are superposed on the slow ones (Gao et al. 2012). The typical duration of the pulses is seconds. Given the typical Lorentz factor ${\rm{\Gamma }}=100{{\rm{\Gamma }}}_{2}$ of GRBs, the typical emission radius is

$$\begin{eqnarray}&&r\sim {{\rm{\Gamma }}}^{2}c\;{t}_{{\rm{pulse}}}\sim (3\times {10}^{14}\;{\rm{cm}}){{\rm{\Gamma }}}_{2}^{2}({t}_{{\rm{pulse}}}/1\;{\rm{s}}).\end{eqnarray} \tag{ 3 }$$

Such a radius is much larger than the photosphere radius, suggesting that the emission occurs in an optically thin region above the photosphere. It is also consistent with the typical input parameters in our previous calculations to interpret the standard Band function spectra of GRBs within the fast-cooling synchrotron radiation model (Uhm & Zhang 2014).

Figure 1 shows an example calculation of the observed spectral fluxes in different energy bands, which are emitted from a relativistic spherical shell. A constant Lorentz factor of ${\rm{\Gamma }}=300$ is adopted, and the emission of the shell is turned on at radius ${r}_{{\rm{on}}}={10}^{14}$ cm. The turn-off radius is taken to be ${r}_{{\rm{off}}}=3\times {10}^{15}$ cm, so that it corresponds to the turn-off time at about ${t}_{{\rm{obs}}}=1.1\;{\rm{s}}$. Note that one can achieve the same turn-off time with different values of ${\rm{\Gamma }},{r}_{{\rm{on}}}$, and ${r}_{{\rm{off}}}$ by simultaneously adjusting them, but our conclusion below still remains unchanged. Beyond the turn-off time ${t}_{{\rm{obs}}}=1.1\;{\rm{s}}$, the observed spectral flux is produced purely by the high-latitude emission. The number of electrons in the shell is assumed to increase at a constant injection rate, ${R}_{{\rm{inj}}}={10}^{47}\;{{\rm{s}}}^{-1}$. The top panels show the light curves¹ in logarithmic scales at four different energies, 30 keV (black), 100 keV (blue), 300 keV (red), and 1 MeV (green), respectively, while the middle panels show the same four light curves in linear scales. The bottom panels show the observed spectra at four different observer times: 0.3 s (black), 1 s (blue), 3 s (red), and 10 s (green), respectively. Hence, the 0.3 s (black) and 1 s (blue) spectra are dominated by the line-of-sight emission, while the 3 s (red) and 10 s (green) spectra are made purely by the high-latitude emission. The four dotted vertical lines in the bottom panels correspond to the energies 30 keV, 100 keV, 300 keV, and 1 MeV, respectively, for which the light curves are calculated.

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We present four different models in Figure 1 with regard to the shape of the photon spectrum in the co-moving frame, as delineated by the functional form H(x). In the first column (model [1a]), H(x) is assumed to have a Band-function shape with the typical observed values ${\alpha }_{{\rm{B}}}=-1$ and ${\beta }_{{\rm{B}}}=-2.3$. We choose² a certain constant value for each ${\nu }_{{\rm{ch}}}^{\prime }$ and ${P}_{0}^{\prime}$, such that the observed spectra are placed at around 1 MeV with a flux density of about 1 mJy, or $2.4\times {10}^{-6}\;\mathrm{erg}\;{\mathrm{cm}}^{-2}\;{{\rm{s}}}^{-1}$, which corresponds to a relatively bright GRB. In the second column (model [1a_i]), we keep everything the same as in model [1a], but adopt ${\alpha }_{{\rm{B}}}=-2/3$ so that the assumed photon spectrum has the same slope as the synchrotron function (Rybicki & Lightman 1979) below the peak area. As one can clearly see from the figure, the light curves of these two models at different energies simultaneously peak at the turn-off time and do not exhibit a spectral lag. Just as predicted from the curvature effect theory of the spherical shell, the 3 s (red) and 10 s (green) spectra (made by the high-latitude emission) are placed at a progressively lower energy than the 0.3 s (black) and 1 s (blue) spectra (dominated by the line-of-sight emission), but with a progressively weaker flux density. For a constant value of Γ, the peak energy E_p and the peak flux density ${F}_{\nu ,p}$ of the observed spectra satisfy the following relation during the high-latitude emission:

$$\begin{eqnarray}&&{F}_{\nu ,p}\propto {E}_{p}^{2},\end{eqnarray} \tag{ 4 }$$

since ${E}_{p}\propto D$ and ${F}_{\nu ,p}\propto {D}^{2}$ (Uhm & Zhang 2015a), where D is the Doppler factor.

In the third column (model [1a_j]), we consider the case where the photon spectrum in the co-moving frame assumes a Planck-function shape, i.e., $H(x)={x}^{3}/({e}^{x}-1)$, and the observed spectra are located at a similar characteristic frequency and flux density. The light curves at different energies still do not exhibit a spectral lag. However, we note that the 1 s (blue), 3 s (red), and 10 s (green) spectra now appear to be coincident with one another asymptotically at low energies. The 3 s (red) and 10 s (green) spectra are expected to coincide with each other at low energies, since $H(x)={x}^{2}$ is satisfied for $x\ll 1$ and the relation ${F}_{\nu ,p}\propto {E}_{p}^{2}$ holds during the high-latitude emission. Given that these two conditions are satisfied, the 1 s (blue) spectrum also appears to be coincident simply because its time 1 s is shortly before the turn-off time at 1.1 s. This coincidence of the spectra indicates a possibility that the 3 s (red) and 10 s (green) spectra can even outshine the 1 s (blue) spectrum at low energies if H(x) assumes a shape harder than $H(x)={x}^{2}$ below the peak area. Thus, in the fourth column (model [1a_k]), we adopt $H(x)={x}^{5}/({e}^{x}-1)$, although unphysical, and place the spectra at a similar characteristic frequency and flux density. As one can see, the 3 s (red) and 10 s (green) spectra indeed outshine the 1 s (blue) spectrum at low energies, and some spectral lags are produced between the light curves at different energies. However, due to the significant suppression of the flux density during the high-latitude emission, the light curves at low energies become too faint. Consequently, the produced spectral lags are essentially not noticeable, as shown in linear scales.³

We conclude that the high-latitude curvature effect cannot produce any spectral lag if the photon spectrum in the co-moving frame takes a shape softer than $H(x)={x}^{2}$ below the peak area. Even if some spectral lags might be possible for a shape harder than this critical case x², the resulting spectral lags are essentially not noticeable.

As the spherical shell expands in a three-dimensional space, the magnetic field strength B in the shell cannot be preserved as a constant; rather, a simple consideration of the flux conservation of the magnetic fields yields $B\propto {r}^{-1}$ for the toroidal component (e.g., Spruit et al. 2001). Also, the possible dissipation of magnetic field energy in the shell (e.g., via reconnection of magnetic field lines) would result in a faster decrease than indicated by the flux conservation. Hence, we consider

$$\begin{eqnarray}&&B{(r)={B}_{0}(r/{r}_{0})}^{-b},\end{eqnarray} \tag{ 6 }$$

where B₀ is a normalization value at radius r₀, and the index $b\geqslant 1$. As we recently showed, this decrease of magnetic field strength in the emitting region is an essential element in reproducing the observed shape of the prompt emission spectra of GRBs (Uhm & Zhang 2014).

Figure 2 shows the observed spectral flux emitted from the spherical shell whose magnetic field strength in the co-moving frame follows Equation (6). As in Figure 1, the shell moves at a constant Lorentz factor ${\rm{\Gamma }}=300$, and its emission is turned on at radius ${r}_{{\rm{on}}}={10}^{14}$ cm. The turn-off radius is taken to be larger than before, ${r}_{{\rm{off}}}=3\times {10}^{16}$ cm, so that it corresponds to a larger turn-off time at about ${t}_{{\rm{obs}}}=11\;{\rm{s}}$. By choosing so, we can focus on the line-of-sight emission for the timescale of seconds, since the high-latitude curvature effect does not play a significant role when ${t}_{{\rm{obs}}}\lt 11\;{\rm{s}}$. The number of electrons in the shell is assumed to increase at a constant injection rate ${R}_{{\rm{inj}}}={10}^{47}\;{{\rm{s}}}^{-1}$. The top panels show the light curves in linear scales at four different energies, 30 keV (black), 100 keV (blue), 300 keV (red), and 1 MeV (green), respectively, while the bottom panels show the observed spectra at four different observer times, 0.3 s (black), 1 s (blue), 3 s (red), and 10 s (green), respectively. Therefore, all four spectra shown here are dominated by the line-of-sight emission. The four dotted vertical lines in the bottom panels correspond to the energies 30 keV, 100 keV, 300 keV, and 1 MeV, respectively, for which the light curves are calculated.

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In Figure 2, we adopt a Band-function shape with the parameters ${\alpha }_{{\rm{B}}}=-0.8$ and ${\beta }_{{\rm{B}}}=-2.3$, as for the shape of the photon spectrum in the co-moving frame. Based on the typical parameters of GRB prompt emission (e.g., Zhang & Yan 2011; Uhm & Zhang 2014), we choose the values ${B}_{0}=30$ G at ${r}_{0}={10}^{15}$ cm and ${\gamma }_{{\rm{ch}}}=8\times {10}^{4}$, which locate the observed spectra at around the observing energy bands. In the first column (model [1b]), we take b = 1 (i.e., flux-conservation case), while in the second column (model [1c]) we take b = 1.25. The third column (model [1d]) takes b = 1.5.

Due to the decreasing strength of the magnetic field in the emitting region, the characteristic frequency ${\nu }_{{\rm{ch}}}^{\prime }$ of the photon spectrum in the co-moving frame, given in Equation (5), decreases in time. Consequently, the observed spectra sweep through the observing energy bands as the observer time ${t}_{{\rm{obs}}}$ elapses. The larger the index b is, the faster the sweep through is. The observed flux-level can also decrease in time, due to a decreasing spectral power ${P}_{0}^{\prime }$ of each electron in the co-moving frame, even though the number of radiating electrons N in the shell increases in time. As one can see from the figure, some spectral lags are produced in our model light curves. We stress that these spectral lags occur well before the turn-off time, i.e., well before the high-latitude emission dominates over the line-of-sight emission.

Although encouraging, the models here with a decreasing strength of magnetic field in the emitting region alone cannot explain the observations. In Section 3.3, we will show that the details of the model light curves shown in Figure 2 are not consistent with the observed properties of spectral lags. In particular, the pulse profiles of these models are too broad at low frequencies, leading to a too-steep slope in the ${t}_{p}\mbox{--}{\nu }_{{\rm{obs}}}$ and width$-{\nu }_{{\rm{obs}}}$ relations.

We conclude this Section by making the important point that in order to produce some spectral lags, the observed spectrum has to be curved and the spectral peak has to move from high-energy to low-energy as a function of time. Imagine that the peak energy of a curved photon spectrum remains constant during an observation time span. Then, moving the spectrum upward or downward would only result in a simultaneous rise or fall in the light curves at different energies, without making any spectral lag.

From observations of the steep-decay phase of GRB X-ray flares (Burrows et al. 2005), we recently found clear observational evidence that the X-ray flare emitting region undergoes rapid bulk acceleration as the photons are emitted, hence providing a signature for a Poynting-flux-dominated outflow in the emitting region of X-ray flares (Uhm & Zhang 2015b). The finding applies to the majority of X-ray flares detected by Swift, suggesting that bulk acceleration may be ubiquitous (Jia et al. 2016). Recalling that the X-ray flares share a similar physical mechanism as the prompt emission itself, we investigate here the effect of bulk acceleration on the formation of spectral lags in prompt emission. We consider

$$\begin{eqnarray}&&{\rm{\Gamma }}{(r)={{\rm{\Gamma }}}_{0}(r/{r}_{0})}^{s},\end{eqnarray} \tag{ 7 }$$

where ${{\rm{\Gamma }}}_{0}$ is a normalization value at radius r₀, and the index s describes the degree of bulk acceleration.

We begin with the models [1b], [1c], and [1d] (presented in Figure 2 for a decreasing profile of B(r) in the emitting region) and let these models undergo bulk acceleration instead of coasting with ${\rm{\Gamma }}(r)=300$. Since no first-principle prediction is available to quantify the acceleration of the jet, we explore different acceleration profiles ${\rm{\Gamma }}(r)$ to reach a satisfactory match with the data. The characteristic electron Lorentz factor ${\gamma }_{{\rm{ch}}}$ is correspondingly adjusted to assure that the observed E_p is around 1 MeV. One example calculation adopts ${{\rm{\Gamma }}}_{0}=250$ at ${r}_{0}={10}^{15}$ cm and s = 0.35, and ${\gamma }_{{\rm{ch}}}=5\times {10}^{4}$, with all other model parameters the same as in Figure 2. The results are shown in Figure 3.

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Shown in the first column in Figure 3 is the model [2b] with b = 1 as for model [1b]. The second column (model [2c]) takes b = 1.25 as in model [1c], while the third column (model [2d]) is with b = 1.5 as in model [1d]. Again, the top panels show the light curves at four different energies, 30 keV (black), 100 keV (blue), 300 keV (red), and 1 MeV (green), respectively, while the bottom panels show the observed spectra at four different observer times, 0.3 s (black), 1 s (blue), 3 s (red), and 10 s (green), respectively. We kept the same turn-on and turn-off radius here as in Figure 2, but for this accelerating profile of ${\rm{\Gamma }}(r)$, the turn-off point corresponds to the turn-off time at about ${t}_{{\rm{obs}}}=4.0\;{\rm{s}}$ in Figure 3. Hence, the 10 s (green) spectra are made purely by the high-latitude emission.

As one can see from the figure, the light curves now exhibit a clear pattern of spectral lags, successfully reproducing the observations. In Section 3.3, we will show that the details of the light curves in models [2c] and [2d] are indeed in a good agreement with the observed properties of spectral lags.

Due to a decreasing profile of B(r) in the emitting region, the observed spectra sweep through the observing energy bands again as the observer time ${t}_{{\rm{obs}}}$ elapses. The larger the index b, the faster the sweep through. However, the sweep through in Figure 3 occurs in a more complicated way than in Figure 2, since the effect of bulk acceleration is added. When an accelerating profile of ${\rm{\Gamma }}(r)$ is combined with a decreasing profile of B(r), the observed flux-level can exhibit an initial rise followed by a fall at a later time. This behavior reshapes our model light curves in such a clever way that their properties come close to observations.

This complicated behavior may be understood by the following rough analytical estimates. For the observed spectra dominated by the line-of-sight emission, we note that the peak energy ${E}_{p}\propto {\rm{\Gamma }}\nu {^{\prime} }_{{\rm{ch}}}\propto {\rm{\Gamma }}B$, whereas the peak flux density ${F}_{\nu ,p}\propto N{\rm{\Gamma }}B$ with the total number of electrons $N\propto t^{\prime} \propto r/{\rm{\Gamma }}$ for constant injection rate in the co-moving frame. Hence, one roughly has

$$\begin{eqnarray}&&{E}_{p}\propto {r}^{s-b}\propto {t}_{{\rm{obs}}}^{\tfrac{s-b}{1-2s}},\;\;{F}_{\nu ,p}\propto {r}^{1-b}\propto {t}_{{\rm{obs}}}^{\tfrac{1-b}{1-2s}},\end{eqnarray} \tag{ 8 }$$

for $s\lt 1/2$. One can see that these two parameters, which are critical to calculate the light curves at different energy bands, both depend on the b and s parameters, but in different ways. Introducing bulk acceleration (s parameter) is essential to reproduce the correct pulse broadness and its energy evolution as observed.

We also remark that the turn off of the shell's emission, visible at about ${t}_{{\rm{obs}}}=4.0\;{\rm{s}}$ in model [2b], is nearly invisible in models [2c] and [2d], so that the high-latitude emission can take over without being noticed. This is also an attractive aspect of the models presented here. We conclude that the combined effect of an accelerating profile of ${\rm{\Gamma }}(r)$ and a decreasing profile of B(r) allows for a successful reproduction of the observed spectral lags, well before the shell's emission is turned off, i.e., without the need to invoke the high-latitude curvature effect.

Now we compare the properties of our model light curves to the observations. We first take model [1b] and find the peak time t_p and the width individually for each of the model's four light curves. The peak time is the observer time where the light curve peaks, and the width is defined as the full observer-time interval that covers half of the maximum flux density of the light curve. In Figure 4, we plot the peak time (top panel) and the width (bottom panel) of each light curve against the observing frequency of that light curve, and then connect the plotted points with a line. We repeat this for all six models [1b], [1c], [1d], [2b], [2c], and [2d], and show the results together in Figure 4 for comparison.

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The two dotted–dashed lines in Figure 4 indicate the relations ${t}_{p}\propto {\nu }_{{\rm{obs}}}^{-1/4}$ (top panel) and ${width}\propto {\nu }_{{\rm{obs}}}^{-0.33}$ (bottom panel), respectively, which represent the observed properties of spectral lags (Norris et al. 1996; Liang et al. 2006). Note that only the slopes of the two relations in log–log scales provide meaningful constraints on the models. Thus, in Figure 4, the two relations are shown with arbitrary normalization and compared with our model results. It is clear that models [1b], [1c], and [1d] are not consistent with the observations, since they have too-steep slopes in at least one of the two relations. Model [2b] is better, but the ${t}_{p}-{\nu }_{{\rm{obs}}}$ slope is still somewhat steeper than the observed relation. On the other hand, models [2c] and [2d] are in a good agreement with the observed properties of spectral lags.